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Thermal vorticity and equilibrium domain

Definition

Let $O$ be a nonempty connected open subset of a smooth time-oriented Lorentzian four-manifold $(M, g)$ of signature $(-, +, +, +)$ , with its Levi-Civita connection $\nabla$ and length-coordinate convention $x^{0} = c t$ . Let $β$ be a smooth future-directed timelike Killing vector field on $O$ , with components in $J^{- 1}$ , and fix $k_{B} > 0$ in ${J K}^{- 1}$ . On $O$ define

$T : = \frac{1}{k_{B} \sqrt{- g (β, β)}}, u : = k_{B} T β, ϖ_{a b} : = - \frac{1}{2} (\nabla_{a} β_{b} - \nabla_{b} β_{a}) .$

Thus $T$ is in kelvins, $g (u, u) = - 1$ , and the covariant two-form $ϖ$ has units $J^{- 1} m^{- 1}$ . For a smooth chemical-potential scalar $μ : O \to R$ in joules, set $ζ : = μ / (k_{B} T)$ and $α : = - ζ$ , both dimensionless. These fields are defined only on the region where $β$ is future timelike; no thermal state or material realization is part of this definition.

Definition references

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Killing fields of a Lorentzian metric

Stationary thermal geometric data

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: The smooth time-oriented Lorentzian domain, its Levi-Civita connection, a smooth future-timelike Killing field $β$ , a smooth scalar $μ$ , and $k_{B} > 0$ as specified in the Definition.

[F1]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: $M$ is a smooth time-oriented four-manifold, $g$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ , timelike means $g (v, v) < 0$ , and future direction is fixed by the time orientation.

[F2]

Killing fields of a Lorentzian metric: a smooth vector field is Killing when $L_{β} g = 0$ .

[F3]

Stationary thermal geometric data: the metric-selected connection is Levi-Civita; the length-coordinate convention is $x^{0} = c t$ , vector $u$ is unit timelike, $β$ has inverse-joule units, and $μ$ is in joules.

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ says that every sequence of nonempty sets has a choice function. It is the inherited foundational assumption of the stationary geometric carrier; this definition uses no stronger choice principle.

Proof

1.1givenF1F3algebra

The scalar $s : = - g (β, β)$ is smooth and strictly positive on $O$ . Hence $s^{- 1 / 2}$ is smooth and positive, so $T = (k_{B} \sqrt{s})^{- 1}$ is a smooth positive kelvin-valued scalar and $u = β / \sqrt{s}$ is a smooth future-directed vector field. Direct contraction gives $g (u, u) = g (β, β) / s = - 1$ .

2.1givenF1F2F3F4algebra∎

Lowering the smooth vector field $β$ with $g$ gives a smooth one-form, and its covariant derivative is a smooth covariant two-tensor. Antisymmetrizing this tensor gives the smooth two-form $ϖ$ ; the inverse-length derivative of the inverse-joule components of $β$ gives its stated units. Since $μ$ and $k_{B} T > 0$ are smooth, $ζ$ and $α = - ζ$ are defined and dimensionless on all of $O$ . The only foundational assumption in this construction is the inherited ${A C}_{ω}$ in [F4]; no selection or stronger choice principle is used.

Thermal vorticity and equilibrium domain

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources